A lower bound for classical Kloosterman sums and an application

We present a lower bound for the classical Kloosterman sum S(a, b; c) where (ab,c) = 1 and c is an odd integer. We apply this lower bound for Kloosterman sums to derive an explicit lower bound in Petersson’s trace formula, subject to a given condition. Consequently, we achieve a modified version of [JS20, Theorem 1.7], where weight k and level N are permitted to vary independently. Using this modified version, we get a lower bound for a weighted trace of the Hecke operator T n acting on the space S k (N), of cusp forms of weight k and level N with (n,N) = 1.